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Mirrors > Home > MPE Home > Th. List > rspsn | Structured version Visualization version GIF version |
Description: Membership in principal ideals is closely related to divisibility. (Contributed by Stefan O'Rear, 3-Jan-2015.) (Revised by Mario Carneiro, 6-May-2015.) |
Ref | Expression |
---|---|
rspsn.b | β’ π΅ = (Baseβπ ) |
rspsn.k | β’ πΎ = (RSpanβπ ) |
rspsn.d | β’ β₯ = (β₯rβπ ) |
Ref | Expression |
---|---|
rspsn | β’ ((π β Ring β§ πΊ β π΅) β (πΎβ{πΊ}) = {π₯ β£ πΊ β₯ π₯}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqcom 2731 | . . . . 5 β’ (π₯ = (π(.rβπ )πΊ) β (π(.rβπ )πΊ) = π₯) | |
2 | 1 | a1i 11 | . . . 4 β’ ((π β Ring β§ πΊ β π΅) β (π₯ = (π(.rβπ )πΊ) β (π(.rβπ )πΊ) = π₯)) |
3 | 2 | rexbidv 3170 | . . 3 β’ ((π β Ring β§ πΊ β π΅) β (βπ β π΅ π₯ = (π(.rβπ )πΊ) β βπ β π΅ (π(.rβπ )πΊ) = π₯)) |
4 | rlmlmod 21055 | . . . 4 β’ (π β Ring β (ringLModβπ ) β LMod) | |
5 | rlmsca2 21051 | . . . . 5 β’ ( I βπ ) = (Scalarβ(ringLModβπ )) | |
6 | baseid 17152 | . . . . . 6 β’ Base = Slot (Baseβndx) | |
7 | rspsn.b | . . . . . 6 β’ π΅ = (Baseβπ ) | |
8 | 6, 7 | strfvi 17128 | . . . . 5 β’ π΅ = (Baseβ( I βπ )) |
9 | rlmbas 21045 | . . . . . 6 β’ (Baseβπ ) = (Baseβ(ringLModβπ )) | |
10 | 7, 9 | eqtri 2752 | . . . . 5 β’ π΅ = (Baseβ(ringLModβπ )) |
11 | rlmvsca 21052 | . . . . 5 β’ (.rβπ ) = ( Β·π β(ringLModβπ )) | |
12 | rspsn.k | . . . . . 6 β’ πΎ = (RSpanβπ ) | |
13 | rspval 21066 | . . . . . 6 β’ (RSpanβπ ) = (LSpanβ(ringLModβπ )) | |
14 | 12, 13 | eqtri 2752 | . . . . 5 β’ πΎ = (LSpanβ(ringLModβπ )) |
15 | 5, 8, 10, 11, 14 | lspsnel 20846 | . . . 4 β’ (((ringLModβπ ) β LMod β§ πΊ β π΅) β (π₯ β (πΎβ{πΊ}) β βπ β π΅ π₯ = (π(.rβπ )πΊ))) |
16 | 4, 15 | sylan 579 | . . 3 β’ ((π β Ring β§ πΊ β π΅) β (π₯ β (πΎβ{πΊ}) β βπ β π΅ π₯ = (π(.rβπ )πΊ))) |
17 | rspsn.d | . . . . 5 β’ β₯ = (β₯rβπ ) | |
18 | eqid 2724 | . . . . 5 β’ (.rβπ ) = (.rβπ ) | |
19 | 7, 17, 18 | dvdsr2 20261 | . . . 4 β’ (πΊ β π΅ β (πΊ β₯ π₯ β βπ β π΅ (π(.rβπ )πΊ) = π₯)) |
20 | 19 | adantl 481 | . . 3 β’ ((π β Ring β§ πΊ β π΅) β (πΊ β₯ π₯ β βπ β π΅ (π(.rβπ )πΊ) = π₯)) |
21 | 3, 16, 20 | 3bitr4d 311 | . 2 β’ ((π β Ring β§ πΊ β π΅) β (π₯ β (πΎβ{πΊ}) β πΊ β₯ π₯)) |
22 | 21 | eqabdv 2859 | 1 β’ ((π β Ring β§ πΊ β π΅) β (πΎβ{πΊ}) = {π₯ β£ πΊ β₯ π₯}) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wb 205 β§ wa 395 = wceq 1533 β wcel 2098 {cab 2701 βwrex 3062 {csn 4621 class class class wbr 5139 I cid 5564 βcfv 6534 (class class class)co 7402 ndxcnx 17131 Basecbs 17149 .rcmulr 17203 Ringcrg 20134 β₯rcdsr 20252 LModclmod 20702 LSpanclspn 20814 ringLModcrglmod 21016 RSpancrsp 21062 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-rep 5276 ax-sep 5290 ax-nul 5297 ax-pow 5354 ax-pr 5418 ax-un 7719 ax-cnex 11163 ax-resscn 11164 ax-1cn 11165 ax-icn 11166 ax-addcl 11167 ax-addrcl 11168 ax-mulcl 11169 ax-mulrcl 11170 ax-mulcom 11171 ax-addass 11172 ax-mulass 11173 ax-distr 11174 ax-i2m1 11175 ax-1ne0 11176 ax-1rid 11177 ax-rnegex 11178 ax-rrecex 11179 ax-cnre 11180 ax-pre-lttri 11181 ax-pre-lttrn 11182 ax-pre-ltadd 11183 ax-pre-mulgt0 11184 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-nel 3039 df-ral 3054 df-rex 3063 df-rmo 3368 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-pss 3960 df-nul 4316 df-if 4522 df-pw 4597 df-sn 4622 df-pr 4624 df-op 4628 df-uni 4901 df-int 4942 df-iun 4990 df-br 5140 df-opab 5202 df-mpt 5223 df-tr 5257 df-id 5565 df-eprel 5571 df-po 5579 df-so 5580 df-fr 5622 df-we 5624 df-xp 5673 df-rel 5674 df-cnv 5675 df-co 5676 df-dm 5677 df-rn 5678 df-res 5679 df-ima 5680 df-pred 6291 df-ord 6358 df-on 6359 df-lim 6360 df-suc 6361 df-iota 6486 df-fun 6536 df-fn 6537 df-f 6538 df-f1 6539 df-fo 6540 df-f1o 6541 df-fv 6542 df-riota 7358 df-ov 7405 df-oprab 7406 df-mpo 7407 df-om 7850 df-1st 7969 df-2nd 7970 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-er 8700 df-en 8937 df-dom 8938 df-sdom 8939 df-pnf 11249 df-mnf 11250 df-xr 11251 df-ltxr 11252 df-le 11253 df-sub 11445 df-neg 11446 df-nn 12212 df-2 12274 df-3 12275 df-4 12276 df-5 12277 df-6 12278 df-7 12279 df-8 12280 df-sets 17102 df-slot 17120 df-ndx 17132 df-base 17150 df-ress 17179 df-plusg 17215 df-mulr 17216 df-sca 17218 df-vsca 17219 df-ip 17220 df-0g 17392 df-mgm 18569 df-sgrp 18648 df-mnd 18664 df-grp 18862 df-minusg 18863 df-sbg 18864 df-subg 19046 df-mgp 20036 df-ur 20083 df-ring 20136 df-dvdsr 20255 df-subrg 20467 df-lmod 20704 df-lss 20775 df-lsp 20815 df-sra 21017 df-rgmod 21018 df-rsp 21064 |
This theorem is referenced by: lidldvgen 21183 zndvds 21433 algextdeglem6 33288 |
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